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MIT2_854F10_prob - Probability Lecturer Stanley B Gershwin...

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Probability Lecturer: Stanley B. Gershwin
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Probability and Trick Question Statistics I flip a coin 100 times, and it shows heads every time. Question: What is the probability that it will show heads on the next flip?
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Probability and Statistics Probability = Statistics Probability: mathematical theory that describes uncertainty. Statistics: set of techniques for extracting useful information from data.
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Interpretations Frequency of probability The probability that the outcome of an experiment is A is P ( A ) if the experiment is performed a large number of times and the fraction of times that the observed outcome is A is P ( A ) .
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Interpretations Parallel universes of probability The probability that the outcome of an experiment is A is P ( A ) if the experiment is performed in each parallel universe and the fraction of universes in which the observed outcome is A is P ( A ) .
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Interpretations Betting odds of probability The probability that the outcome of an experiment is A is P ( A ) if before the experiment is performed a risk-neutral observer would be willing to bet $1 against more than $ 1 P ( A ) . P ( A )
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Interpretations State of belief of probability The probability that the outcome of an experiment is A is P ( A ) if that is the opinion (ie, belief or state of mind) of an observer before the experiment is performed.
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Interpretations Abstract measure of probability The probability that the outcome of an experiment is A is P ( A ) if P () satisfies a certain set of conditions: the axioms of probability.
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Interpretations Abstract measure of probability Axioms of probability Let U be a set of samples . Let E 1 , E 2 , ... be subsets of U . Let φ be the null set (the set that has no elements). 0 P ( E i ) 1 P ( U ) = 1 P ( φ ) = 0 If E i E j = φ , then P ( E i E j ) = P ( E i ) + P ( E j )
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Probability Discrete Sample Space Basics Subsets of U are called events. P ( E ) is the probability of E .
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Probability Discrete Sample Space Basics U Low probability High probability
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Probability Set Theory Basics Venn diagrams U A A P ( A ) = 1 P ( A ) ¯
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Probability Set Theory Basics Venn diagrams U AUB U A B A B P ( A B ) = P ( A ) + P ( B ) P ( A B )
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Probability Independence Basics A and B are independent if P ( A B ) = P ( A ) P ( B ) .
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Probability Conditional Probability Basics If P ( B ) = 0 , P ( A | B ) = P ( A B ) P ( B ) U AUB U A B A B We can also write P ( A B ) = P ( A | B ) P ( B ) .
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Probability Conditional Probability Basics Example Throw a die. A is the event of getting an odd number (1, 3, 5). B is the event of getting a number less than or equal to 3 (1, 2, 3). Then P ( A ) = P ( B ) = 1 / 2 and P ( A B ) = P (1 , 3) = 1 / 3 . Also, P ( A | B ) = P ( A B ) /P ( B ) = 2 / 3 .
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Probability Law of Total Probability Basics U C U U B A D A C A D Let B = C D and assume C D = φ . Then P ( A | C ) = P ( A C ) and P ( A | D ) = P ( A D ) . P ( C ) P ( D )
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Probability Law of Total Probability Basics Also, P ( C | B ) = P ( C B ) = P ( C ) because C B = C .
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